ARTICLE IN PRESS
Journal of Fluids and Structures 22 (2006) 345369 www.elsevier.com/locate/jfs
Flow-induced vibration of a square cylinder without and with interferenceR. Ajith Kumara,, B.H.L. Gowdaba
Department of Mechanical Engineering, Amrita Vishwa VidyaPeetham (Deemed University), Ettimadai Campus, Coimbatore 641 105 Tamil Nadu, India b Department of Applied Mechanics, Indian Institute of Technology Madras, Chennai 36, India Received 2 April 2004; accepted 20 November 2005 Available online 10 February 2006
Abstract This paper presents the results of an investigation on the interference effects of a rigid square cylinder on the transverse vibrations of a spring-mounted square cylinder (test cylinder) exposed to a uniform ow. The interference effects were studied for the tandem, side-by-side and staggered arrangements. Experiments have been carried out for various relative dimensions of the test cylinder and the interfering cylinder; the tests for the staggered arrangements were conducted at several tandem distances between the two. The results indicate that there is a critical combination of relative dimensions and spacing that gives rise to maximum amplitude of vibration. Among the cases studied, tandem arrangement with L=B 1:25 and b=B 0:5 gives rise to maximum amplitude of vibration with a=Bmax 0:57. A tentative explanation is offered for the observed features based on ow-visualization studies conducted as a part of the experimental investigation. r 2006 Elsevier Ltd. All rights reserved.
1. Introduction When a exibly mounted body, either streamlined (e.g., an aerofoil) or bluff (e.g., a circular or square cylinder), is exposed to a steady and uniform ow, uidstructure interactions can take place as a result of which the body can experience uctuating pressure forces of considerable magnitude causing the body to undergo induced vibrations. Another body or bodies in its vicinity (situated in the ow eld) can be expected to alter the magnitude and phasing of pressure uctuations acting on the body and thereby the induced vibrations. Aerodynamic interference between bodies is found to depend on various factors such as the body geometry, reduced velocity, longitudinal and transverse spacing between them (and hence, on the arrangement) and also on their support conditions [e.g., Bokaian and Geoola (1985), Zdravkovich (1985), Takeuchi (1990), Takeuchi and Matsumoto (1993)]. Literature shows that considerable amount of data are available on the ow-induced oscillations of bodies with circular cross-section, whereas it is found to be limited on prismatic bodies with square and rectangular cross-sections. This is true for the cases with body in isolation and with interference. Interference effects on the ow-induced vibrations of cylindrical bodies with circular cross-section and prismatic bodies with square cross-section are important in many
Corresponding author. Tel.: +91 422 2656422x362; fax: +91 422 2656274.
E-mail address: email@example.com (R.A. Kumar). 0889-9746/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.juidstructs.2005.11.006
ARTICLE IN PRESS346 R.A. Kumar, B.H.L. Gowda / Journal of Fluids and Structures 22 (2006) 345369
Nomenclature a a0 B b D d f peak-to-peak amplitude distance of separation of the test cylinder from the mean position side dimension of the test cylinder side dimension of the interfering cylinder diameter of the test circular cylinder diameter of the interfering circular cylinder fundamental natural frequency of the springcylinder system
ks L m T U d r
Scruton number: mass damping parameter (2 md/pB2) longitudinal spacing between axes of cylinders mass per unit length of the test cylinder transverse spacing between axes of cylinders free stream velocity logarithmic decrement density of air
practical situations such as ow around chimney stacks, tube bundles in heat exchangers, power transmission lines, offshore structures, bridge pylons, building structures, etc. On square cylinders, for the single cylinder case there are enough data, but comparatively less information on cases with interference. The investigations of Vickery (1966), Bostok and Mair (1972), Otsuki et al. (1974), Nakamura and Mizota (1975), Wilkinson (1981), Bearman and Obasaju (1982), Olivari (1983), Sakamoto (1985) and Bearman and Luo (1988) have dealt with single square cylinder cases wherein they have brought out various aerodynamic characteristics and forces acting on a square cylinder. Interference effects between two square cylinders in tandem arrangement were investigated by Blessmann and Riera (1985), Shiraishi et al. (1986), Sakamoto et al. (1987), Luo and Teng (1990) and Takeuchi and Matsumoto (1992, 1993) wherein they have brought out various aerodynamic properties, response characteristics and mechanisms underlying aerodynamic behaviour of square cylinders. Similar aerodynamic characteristics were brought out for square cylinders in side-by-side and staggered arrangements by other investigators (Bailey and Kwok, 1985; Sakamoto and Haniu, 1988; Taniike and Inaoka, 1988; Taniike, 1992). In the present study, the vibratory response of a spring-mounted square cylinder (test cylinder) due to the interference of another rigid square cylinder (interfering cylinder) placed in tandem, side-by-side and staggered arrangements is described. The results for the test cylinder without interference are rst obtained. Then, the interference studies have been carried out at a velocity corresponding to the peak amplitude of the isolated cylinder without interference. In all the cases considered, the interfering cylinder was never upstream of the test cylinder. In an attempt to bring out the inuence of geometry, in some of the cases, the results are compared with those of circular geometry (Sreedharan, 1992; Gowda and Sreedharan, 1994). The response of the test cylinder was obtained for different ratios of interfering cylinder side dimension b to the test cylinder side dimension B: i.e., b=B 0:5, 1.0, 1.5 and 2.0 (see Fig. 1).
2. Experimental set-up 2.1. Wind tunnel facility The arrangement is essentially the same as that used by Gowda and Deshkulkarni (1988) and Gowda and Prabhu (1987), Gowda and Sreedharan, (1994). The experiments were conducted on an aluminium tube with a square crosssection of side 12 mm, wall thickness 1 mm and length 140 mm. The cylinder was positioned vertically at the centre of a rectangular frame supported by four springs. The frame was positioned in front of an open-circuit wind tunnel with a square exit duct measuring 120 mm 120 mm such that the cylinder was at a distance of 44 mm from the exit of the tunnel. The velocity is uniform over the 70% of the exit cross-section (variation is less than 1%) and reduces gradually towards the edges. One of the springs supporting the cylinder was connected to a dynamic pick up (Type 8001; Bruel and Kjaer). The pick up was in turn connected to a storage oscilloscope (Type 1744A; Hewlett Packard) through a charge amplier (Type 2626; Bruel and Kjaer). The cylinder was capable of vibrating in a direction transverse to the oncoming ow. The natural frequency and the logarithmic decrement of the spring-mounted cylinder were determined to be 54 Hz and 0.00368, respectively. The mass damping parameter ks (Scruton number) was equal to 3.2 (see Nomenclature). The interfering cylinders used for tandem and staggered arrangement were made out of solid aluminium rods with a smooth nish and square cross-section. They were each of length 140 mm and side dimension ratios (b/B) of 0.5, 1.0, 1.5
ARTICLE IN PRESSR.A. Kumar, B.H.L. Gowda / Journal of Fluids and Structures 22 (2006) 345369 347
Fig. 1. The conguration tested. Cases studied: b=B 0:5, 1.0, 1.5 and 2.0.
and 2.0 were used. For the side-by-side arrangement, a different set of cylinders of the same relative dimensions as for the tandem and staggered arrangements were used. These cylinders were 120 mm long and could be conveniently placed besides the test cylinder. Further details of the wind tunnel test set-up have been given by Gowda and Prabhu (1987). However, the test set-up is given here for the sake of clarity, in Figs. 2(a)(d). The response of the test cylinder without any interference was rst determined. Then, the interference studies were carried out by considering various interference cases constituting tandem, side-by-side and staggered arrangements. The range of Reynolds number referred to the test cylinder side dimension is between 3000 and 11 000.
2.2. Flow visualization facility The ow-visualization experiments are carried out for two situations: (a) with two stationary cylinders and (b) with one cylinder oscillated and the other cylinder stationary. These are carried out in a recirculating water channel shown in Fig. 3. It consists of a tank 2.5 m 1.5 m with a depth of 150 mm, at one end of which are located two sets of aluminium discs (vanes) with suitable spacing between them. When these vanes are rotated, they act as paddles and create a ow that is suitably guided to the test-section where the models are placed. A variable speed DC motor is used for rotating the vanes and a fairly wide range of ow speeds can be achieved in the test-section. The uid in the tank is water and ne aluminium powder is used as tracer medium. In situation (a), corresponding to the test cylinder in air experiments, the cylinder employed is a square rod with side dimension 30 mm. Interfering cylinders are also square rods with b=B 0:5, 1.0, 1.5 and 2.0. Experiments are conducted at a Reynolds number of 5200. This is the Reynolds number corresponding to the velocity at the maximum amplitude of the test cylinder under isolated condition in air experiments. The spacing ratios L/B and T/B are systematically varied for each value of b/B and the ow patterns